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Beliebt Rechnen >

integral of 1/(kx-nx^2)

Lösung

\int \frac{1}{k x - n x ^{2}} d x

Lösung

\frac{1}{k} (ln (\frac{∣ n ∣ ∣ x ∣}{∣ k ∣}) - ln \frac{n x}{k} - 1) + C

Schritte zur Lösung

\int \frac{1}{k x - n x ^{2}} d x

Faktorisiere

\frac{1}{k x - n x ^{2}}

= \int \frac{1}{x ( k - n x )} d x

k - n x = k (1 - \frac{n x}{k})

= \int \frac{1}{x k ( 1 - \frac{n x}{k} )} d x

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{k} \cdot \int \frac{1}{x ( 1 - \frac{n x}{k} )} d x

Wende U-Substitution an

= \frac{1}{k} \cdot \int \frac{1}{u ( - u + 1 )} d u

Ermittle den Partialbruch von

\frac{1}{u ( - u + 1 )} : \frac{1}{u} - \frac{1}{u - 1}

= \frac{1}{k} \cdot \int \frac{1}{u} - \frac{1}{u - 1} d u

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{k} (\int \frac{1}{u} d u - \int \frac{1}{u - 1} d u)

\int \frac{1}{u} d u = ln ∣ u ∣

\int \frac{1}{u - 1} d u = ln ∣ u - 1 ∣

= \frac{1}{k} (ln ∣ u ∣ - ln ∣ u - 1 ∣)

Setze in

u = \frac{n x}{k}

ein

= \frac{1}{k} (ln \frac{n x}{k} - ln \frac{n x}{k} - 1)

Vereinfache

\frac{n x}{k} : \frac{∣ n ∣ ∣ x ∣}{∣ k ∣}

= \frac{1}{k} (ln (\frac{∣ n ∣ ∣ x ∣}{∣ k ∣}) - ln \frac{n x}{k} - 1)

Füge eine Konstante zur Lösung hinzu

= \frac{1}{k} (ln (\frac{∣ n ∣ ∣ x ∣}{∣ k ∣}) - ln \frac{n x}{k} - 1) + C

Graph

Beliebte Beispiele

steigung (1.4)(-5)

s l o p e (1.4) (- 5)

integral of 6(sin(2x))/(sin(x))

\int 6 \frac{sin ( 2 x )}{sin ( x )} d x

integral of 1+3x+18x^2

\int 1 + 3 x + 18 x^{2} d x

limit as x approaches 0+of (ln(x)+1)/x

x \to 0 + lim (\frac{ln ( x ) + 1}{x})

inverselaplace (3(s+1))/(s^2+1/2)

in v erse l a pl a ce \frac{3 ( s + 1 )}{s ^{2} + \frac{1}{2}}